Coburn J.W. Algebra and Trigonometry
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580 CHAPTER 5 An Introduction to Trigonometric Functions 5-78
Solution
Using the primary interval and the asymptotes at and we find the
period is To find the value of B we substitute for P in
and find (verify). This gives the equation
To find A, we take the point shown, and use with to
solve for A:
substitute for B
substitute for y and for t
multiply
solve for A
result
The equation of the graph is
Now try Exercises 41 through 46
E. Applications of Tangent and Cotangent Functions
We end this section with one example of how tangent and cotangent functions can be
applied. Numerous others can be found in the exercise set.
EXAMPLE 7
Applications of : Modeling the Movement of a Light Beam
One evening, in port during a Semester at Sea,
Richard is debating a project choice for his
Precalculus class. Looking out his porthole, he
notices a revolving light turning at a constant
speed near the corner of a long warehouse. The
light throws its beam along the length of the
warehouse, then disappears into the air, and then
returns time and time again. Suddenly—Richard
has his project. He notes the time it takes the
beam to traverse the warehouse wall is very
close to 4 sec, and in the morning he measures
the wall’s length at 127.26 m. His project? Modeling the distance of the beam from
the corner of the warehouse as a function of time using a tangent function.
Can you help?
Solution
The equation model will have the form where D(t) is the distance
(in meters) of the beam from the corner after t sec. The distance along the wall
is measured in positive values so we’re using only the period of the function,
giving (the beam “disappears” at ) so Substitution in the period
formula gives and the equation D A tana
8
tb.B
8
P 8.t 4
1
2
P 4
1
2
D1t2 A tan1Bt2,
y A tan1Bt2
y 2 tan1
3
2
t2.
2
A
2
tana
3
4
b
2 A tana
3
4
b
2
2 2 A tanca
3
2
ba
2
bd
3
2
y A tana
3
2
tb
y 2t
2
a
2
, 2b
y A tan
a
3
2
tb.B
3
2
P
B
2
3
P
3
a
3
b
2
3
.
t
3
,t
3
D. You’ve just learned how
to graph and
with various
values of A and B
y A cot1Bt2
y A tan1Bt2
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5-79 Section 5.6 Graphs of Tangent and Cotangent Functions 581
Knowing the beam travels 127.26 m in about 4 sec (when it disappears into
infinity), we’ll use and D 127.26 in order to solve for A and complete
our equation model (see note following this example).
equation model
substitute 127.26 for D and 3.9 for t
solve for A
result
One equation approximating the distance of the beam from the corner of the
warehouse is
Now try Exercises 49 through 52
For Example 7, we should note the choice of 3.9 for t was arbitrary, and while we
obtained an “acceptable” model, different values of A would be generated for other
choices. For instance, gives while gives The true
value of A depends on the distance of the light from the corner of the warehouse wall.
In any case, it’s interesting to note that at sec (one-half the time it takes the beam
to disappear), the beam has traveled only 5m from the corner of the building:
Although the light is rotating at a constant angular speed,
the speed of the beam along the wall increases dramatically as t gets close to 4 sec.
D122 5 tana
4
b 5 m.
t 2
A 0.5.t 3.99A 2.5,t 3.95
D1t2 5 tana
8
tb.
5
A
127.26
tanc
8
13.92d
A tanc
8
13.92d 127.26
A tana
8
tb D
t 3.9
E. You’ve just learned how
to solve applications of
and y cot ty tan t
Zeroes, Asymptotes, and the Tangent/Cotangent Functions
TECHNOLOGY HIGHLIGHT
In this Technology Highlight we’ll explore the tangent and cotangent
functions from the perspective of their ratio definition. While we could
easily use to generate and explore the graph, we would
miss an opportunity to note the many important connections
that emerge from a ratio definition perspective. To begin, enter
and as shown in Figure 5.76 [recall
that function variables are accessed using
(Y-VARS) (1:Function)]. Note that Y
2
has been
disabled by overlaying the cursor on the equal sign and
pressing . In addition, note the slash next to Y
1
is more
bold than the other slashes. The TI-84 Plus offers options that
help distinguish between graphs when more than one is being
displayed, and we selected a bold line for Y
1
by moving the
cursor to the far left position and repeatedly pressing
until the desired option appeared. Pressing
7:ZTrig at this point produces the screen shown in Figure 5.77,
where we note that tan x is zero everywhere that sin x
ZOOM
ENTER
ENTER
ENTER
VARS
Y
3
Y
1
Y
2
,Y
1
sin x, Y
2
cos x,
Y
1
tan x
Figure 5.76
6.2
4
4
6.2
Figure 5.77
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582 CHAPTER 5 An Introduction to Trigonometric Functions 5-80
is zero. This is hardly surprising since but is a
point that is often overlooked. Going back to the screen
and disabling Y
1
while enabling Y
2
will produce the graph
shown in Figure 5.78.
Exercise 1: What do you notice about the zeroes of cos x as
they relate to the graph of ?
Exercise 2: Go to the screen and change Y
3
from
(tangent) to (cotangent), then repeat the
previous investigation regarding and
.
y cos xy sin x
Y
2
Y
1
Y
1
Y
2
Y =
Y
3
tan x
Y =
tan x
sin x
cos x
,
6.2
4
4
6.2
Figure 5.78
5.6 EXERCISES
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase.
Carefully reread the section if needed.
1. The period of and is ________.
To find the period of and
the formula _________ is used.
2. The function is ___________ everywhere
it is defined. The function is
___________ everywhere it is defined.
3. Tan t and cot t are _______ functions, so
___________. If then
_________.tana
11
12
b
0.268,tana
11
12
b,
f 1t2
y cot t
y tan t
y cot1Bt2,y tan1Bt2
y cot ty tan t
4. The asymptotes of _________ are located at
odd multiples of The asymptotes of
_________ are located at integer multiples of
5. Discuss/Explain how you can obtain a table of
values for (a) given the values for
and and (b) given the values
for
6. Explain/Discuss how the zeroes of and
are related to the graphs of and
How can these relationships help graph
functions of the form and
?y A cot1Bt2
y A tan1Bt2
y cot t.
y tan ty cos t
y sin t
y tan t.
y cos t,y sin t
y cot t
.
y
2
.
y
DEVELOPING YOUR SKILLS
Use the values given for sin t and cos t to complete the tables.
7.
8.
t
0
0
tan t
y
x
1
2
12
2
13
2
1cos t x
1
13
2
12
2
1
2
sin t y
3
2
4
3
5
4
7
6
0
01
tan t
y
x
13
2
12
2
1
2
cos t x
1
2
12
2
13
2
1sin t y
2
11
6
7
4
5
3
3
2
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9. Without reference to a text or calculator, attempt to
name the decimal equivalent of the following
values to one decimal place.
10. Without reference to a text or calculator, attempt to
name the decimal equivalent of the following
values to one decimal place.
11. State the value of each expression without the use
of a calculator.
a. b.
c. d.
12. State the value of each expression without the use
of a calculator.
a. b.
c. d.
13. State the value of t without the use of a calculator,
given terminates in the quadrant
indicated.
a. in QIV
b. in QIII
c. in QIV
d. in QII
14. State the value of t without the use of a calculator,
given terminates in the quadrant
indicated.
a. in QI
b. in QII
c. in QI
d. in QIIIcot t 1, t
tan t
1
13
, t
tan t 13
, t
cot t 1, t
t 30, 22
tan t 1, t
cot t
1
13
, t
cot t 13
, t
tan t 1, t
t 30, 22
cota
5
6
btana
5
4
b
tan cota
2
b
tana
3
bcota
3
4
b
cota
6
btana
4
b
1
13
13
2
13
3
2
3
2
13
12
2
12
6
4
2
Use the values given for sin t and cos t to complete the
tables.
15.
16.
17. Given is a solution to use the
period of the function to name three additional
solutions. Check your answer using a calculator.
18. Given is a solution to use the
period of the function to name three additional
solutions. Check your answer using a calculator.
19. Given is a solution to use the
period of the function to name three additional
solutions. Check your answers using a calculator.
20. Given is a solution to use the
period of the function to name three additional
solutions. Check your answers using a calculator.
Verify the value shown for t is a solution to the equation
given, then use the period of the function to name all
real roots. Check two of these roots on a calculator.
21.
22.
23.
24. t
5
12
; cot t 2 13
t
12
; cot t 2 13
t
16
; tan t 0.1989
t
10
; tan t 0.3249
tan t 3,t 1.25
cot t 0.07,t 1.5
cot t 0.77,t
7
24
tan t 7.6,t
11
24
5-81 Section 5.6 Graphs of Tangent and Cotangent Functions 583
t
0
0
cot t
x
y
1
2
12
2
13
2
1cos t x
1
13
2
12
2
1
2
sin t y
3
2
4
3
5
4
7
6
0
01
cot t
x
y
13
2
12
2
1
2
cos t x
1
2
12
2
13
2
1sin t y
2
11
6
7
4
5
3
3
2
College Algebra & Trignometry—
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Graph each function over the interval indicated, noting
the period, asymptotes, zeroes, and value of A. Include a
comparative sketch of or as
indicated.
25.
26.
27.
28.
Graph each function over the interval indicated, noting
the period, asymptotes, zeroes, and value of A and B.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
p1t2
1
2
cota
4
tb; 34, 44
f 1t2 2 cot1t2; 31, 14
y 4 tana
2
tb; 32, 24
y 3 tan12t2; c
1
2
,
1
2
d
y
1
2
cot12t2; c
2
,
2
d
y 5 cota
1
3
tb; 33, 34
y 4 tana
1
2
tb; 32, 24
y 2 tan14t2; c
4
,
4
d
y cota
1
2
tb; 32, 24
y cot14t2; c
4
,
4
d
y tana
1
4
tb; 34, 44
y tan12t2; c
2
,
2
d
r1t2
1
4
cot t; 32, 24
h1t2 3 cot t; 32, 24
g1t2
1
2
tan t; 32, 24
f 1t2 2 tan t; 32, 24
y cot ty tan t
Find the equation of each graph, given it is of the form
41.
42.
Find the equation of each graph, given it is of the form
.
43.
44.
45. Given that and are solutions to
use a graphing calculator to find
two additional solutions in
46. Given is a solution to use
a graphing calculator to find two additional
solutions in
31, 14.
tan12t2 cot1t2,t
1
6
30, 24.
cot13t2 tan t,
t
3
8
t
8
1
9
,
√3
1
6
1
2
1
3
1
6
1
3
1
2
y
t
3
3
1
4
,
2√3
y
t
3
3
3
2123
1
y A cot1Bt2
y
t
6
6
3
3
2
2
2
1
2
1
12
1
2
,
y
t
3
2
2
2
2
9
9
2
,
3
y A tan1Bt2.
584
CHAPTER 5 An Introduction to Trigonometric Functions 5-82
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5-83 Section 5.6 Graphs of Tangent and Cotangent Functions 585
47. The height of an object calculated from a
distance:
The height h of a tall
structure can be
computed using two
angles of elevation
measured some
distance apart along a
straight line with the
object. This height is
given by the formula shown, where d is the
distance between the two points from which angles
u and v were measured. Find the height h of a
building if and
d 100 ft.u 40°, v 65°,
dx d
h
uv
h
d
cot u cot v
48. Position of an image reflected from a spherical
lens:
The equation
shown is used to
help locate the
position of an
image reflected
by a spherical
mirror, where s
is the distance of
the object from
the lens along a horizontal axis, is the angle of
elevation from this axis, h is the altitude of the
right triangle indicated, and k is distance from the
lens to the foot of altitude h. Find the distance k
given , and that the object is
24 mm from the lens.
24
,h 3 mm
tan
h
s k
WORKING WITH FORMULAS
Tangent function data models: Model the data in
Exercises 49 and 50 using the function tan(Bx).
State the period of the function, the location of the
asymptotes, the value of A, and name the point (x, y) used
to calculate A (answers may vary). Use your equation
model to evaluate the function at and What
observations can you make? Also see Exercise 58.
49.
50.
x 2.x 2
y A
51. As part of a lab setup, a
laser pen is made to
swivel on a large
protractor as illustrated in
the figure. For their lab
project, students are
asked to take the
instrument to one end of
a long hallway and
measure the distance of
the projected beam
relative to the angle the
pen is being held, and
collect the data in a
table. Use the data to
find a function of the
form
State the period of the
function, the location of
the asymptotes, the value
of A, and name the point
() you used to calculate A (answers may vary).
Based on the result, can you approximate the
length of the laser pen? Note that in degrees, the
period formula for tangent is
52. Use the equation model obtained in Exercise 51 to
compare the values given by the equation with the
actual data. As a percentage, what was the largest
deviation between the two?
P
180°
B
.
, y
y A tan1B2.
APPLICATIONS
Lens
P
Object
P
Reflected
image
k
s
h
Input Output Input Output
1 1.4
23
3 5.2
4 9.7
520
6
00
q1.41
32
5.23
9.74
205
q6
Input Output Input Output
0.5 6.4
1 13.7
1.5 23.7
2 44.3
2.5 91.3
3
00
q6.40.5
13.71
23.71.5
44.32
91.32.5
q3
Laser Light
Distance
(degrees) (cm)
00
10 2.1
20 4.4
30 6.9
40 10.1
50 14.3
60 20.8
70 33.0
80 68.1
89 687.5
College Algebra & Trignometry—
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53. Circumscribed polygons:
The perimeter of a regular
polygon circumscribed
about a circle of radius r is
given by
where n is the number of
sides and r is the
radius of the circle. Given
cm, (a) What is the
circumference of the circle? (b) What is
the circumference of the polygon when ?
Why? (c) Calculate the perimeter of the polygon
for and 100. What do you notice?
54. Circumscribed polygons: The area of a regular
polygon circumscribed about a circle of radius r is
given by where n is the number
of sides and r is the radius of the circle.
Given cm,
a. What is the area of the circle?
b. What is the area of the polygon when ?
Why?
c. Calculate the area of the polygon for
and 100. What do you notice?
Coefficients of friction:
Pulling someone on a
sled is much easier
during the winter than
in the summer, due to
a phenomenon known
as the coefficient of
friction. The friction
between the sled’s skids
and the snow is much lower than the friction between the
skids and the dry ground or pavement. Basically, the
coefficient of friction is defined by the relationship
m tan where is the angle at which a block composed
of one material will slide down an inclined plane made of
another material, with a constant velocity. Coefficients of
friction have been established experimentally for many
materials and a short list is shown here.
55. Graph the function with in degrees
over the interval and use the graph to
estimate solutions to the following. Confirm or
contradict your estimates using a calculator.
a. A block of copper is placed on a sheet of steel,
which is slowly inclined. Is the block of copper
moving when the angle of inclination is ?
At what angle of inclination will the copper
block be moving with a constant velocity down
the incline?
30°
30°, 60°4
tan ,
,
n 10, 20, 30,
n 4
r 10
1n 32
A nr
2
tana
n
b,
n 10, 20, 30,
n 4
r 10
1n 32
P 2nr tana
n
b,
b. A block of copper is placed on a sheet of cast-
iron. As the cast-iron sheet is slowly inclined,
the copper block begins sliding at a constant
velocity when the angle of inclination is
approximately What is the coefficient of
friction for copper on cast-iron?
c. Why do you suppose coefficients of friction
greater than are extremely rare? Give
an example of two materials that likely have a
high m-value.
56. Graph the function with in radians
over the interval and use the graph to
estimate solutions to the following. Confirm or
contradict your estimates using a calculator.
a. A block of glass is placed on a sheet of glass,
which is slowly inclined. Is the block of glass
moving when the angle of inclination is ?
What is the smallest angle of inclination for
which the glass block will be moving with a
constant velocity down the incline (rounded to
four decimal places)?
b. A block of Teflon is placed on a sheet of steel.
As the steel sheet is slowly inclined, the Teflon
block begins sliding at a constant velocity
when the angle of inclination is approximately
0.04. What is the coefficient of friction for
Teflon on steel?
c. Why do you suppose coefficients of friction
less than are extremely rare for two
solid materials? Give an example of two
materials that likely have a very low m value.
57. Tangent lines: The actual
definition of the word
tangent comes from the
Latin tangere, meaning “to
touch.” In mathematics, a
tangent line touches the
graph of a circle at only one
point and function values for
are obtained from the
length of the line segment tangent to a unit circle.
a. What is the length of the line segment when
?
b. If the line segment is 16.35 units long, what is
the value of ?
c. Can the line segment ever be greater than
100 units long? Why or why not?
d. How does your answer to (c) relate to the
asymptotic behavior of the graph?
80°
tan
1
tan
0.04
4
c0,
5
12
d
tan
2.5
46.5°.
586 CHAPTER 5 An Introduction to Trigonometric Functions 5-84
r
Material Coefficient
steel on steel 0.74
copper on glass 0.53
glass on glass 0.94
copper on steel 0.68
wood on wood 0.5
College Algebra & Trignometry—
Exercise 53
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5-85 Section 5.7 Transformations and Applications of Trigonometric Graphs 587
60. (5.1) A lune is a
section of surface area
on a sphere, which is
subtended by an
angle at the
circumference. For
in radians, the surface
area of a lune is
where r is
the radius of the
sphere. Find the area
of a lune on the surface of the Earth which is
subtended by an angle of Assume the radius of
the Earth is 6373 km.
61. (3.4/3.5) Find the y-intercept, x-intercept(s), and
all asymptotes of each function, but do not graph.
a. b.
t1x2
x 1
x
2
4x
h1x2
3x
2
9x
2x
2
8
15°.
A 2r
2
,
c.
62. (5.2) State the points on the unit circle that
correspond to and
What is the value of ? Why?
63. (4.1) The radioactive element potassium-42 is
sometimes used as a tracer in certain biological
experiments, and its decay can be modeled by the
formula , where Q(t) is the
amount that remains after t hours. If 15 grams (g)
of potassium-42 are initially present, how many
hours until only 10 g remain?
Q1t2 Q
0
e
0.055t
tana
2
b
2.t 0,
4
,
2
, ,
3
4
,
3
2
,
p1x2
x
2
1
x 2
MAINTAINING YOUR SKILLS
r
From your algebra experience, you may remember beginning with a study of linear
graphs, then moving on to quadratic graphs and their characteristics. By combining
and extending the knowledge you gained, you were able to investigate and understand
a variety of polynomial graphs—along with some powerful applications. A study of
trigonometry follows a similar pattern, and by “combining and extending” our under-
standing of the basic trig graphs, we’ll look at some powerful applications in this
section.
5.7 Transformations and Applications of Trigonometric Graphs
Learning Objectives
In Section 5.7 you will learn how to:
A. Apply vertical
translations in context
B. Apply horizontal
translations in context
C. Solve applications
involving harmonic
motion
58. Rework Exercises 49 and 50, obtaining a new
equation for the data using a different ordered pair
to compute the value of A. What do you notice?
Try yet another ordered pair and calculate A once
again for another equation Y
2
. Complete a table of
values using the given inputs, with the outputs of
the three equations generated (original, Y
1
, and Y
2
).
Does any one equation seem to model the data
better than the others? Are all of the equation
models “acceptable”? Please comment.
59. Regarding Example 7, we can use the standard
distance/rate/time formula to compute theD RT
average velocity of the beam of light along the wall
in any interval of time: For example, using
the average velocity in the
interval [0, 2] is
Calculate the average velocity of the beam in the
time intervals [2, 3], [3, 3.5], and [3.5, 3.8] sec.
What do you notice? How would the average
velocity of the beam in the interval [3.9, 3.99] sec
compare?
D122 D102
2 0
2.5 m/sec.
D1t2 5 tana
8
tb,
R
D
T
.
EXTENDING THE CONCEPT
College Algebra & Trignometry—
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588 CHAPTER 5 An Introduction to Trigonometric Functions 5-86
A. Vertical Translations:
On any given day, outdoor temperatures tend to follow a sinusoidal pattern, or a
pattern that can be modeled by a sine function. As the sun rises, the morning
temperature begins to warm and rise until reaching its high in the late afternoon,
then begins to cool during the early evening and nighttime hours until falling to
its nighttime low just prior to sunrise. Next morning, the cycle begins again. In the
northern latitudes where the winters are very cold, it’s not unreasonable to assume
an average daily temperature of and a temperature graph in degrees
Celsius that looks like the one in Figure 5.79. For the moment, we’ll assume that
corresponds to 12:00 noon. Note that and yielding
or
If you live in a more temperate area, the daily temperatures still follow a sinusoidal
pattern, but the average temperature could be much higher. This is an example of a
vertical shift, and is the role D plays in the equation All other
aspects of a graph remain the same; it is simply shifted D units up if and D
units down if As in Section 5.3, for maximum value M and minimum value m,
gives the amplitude A of a sine curve, while gives the average value D.
EXAMPLE 1
Modeling Temperature Using a Sine Function
On a fine day in Galveston, Texas, the high temperature might be about with
an overnight low of
a. Find a sinusoidal equation model for the daily temperature.
b. Sketch the graph.
c. Approximate what time(s) of day the temperature is Assume
corresponds to 12:00 noon.
Solution
a. We first note the period is still so and the equation model
will have the form Using we find
the average value with amplitude The resulting
equation is
b. To sketch the graph, use a reference rectangle units tall and
units wide, along with the rule of fourths to locate zeroes and max/min values
(see Figure 5.80). Then lightly sketch a sine curve through these points and
within the rectangle as shown. This is the graph of
Using an appropriate scale, shift the rectangle and plotted points vertically
upward 73 units and carefully draw the finished graph through the points and
within the rectangle (see Figure 5.81).
y 12 sina
12
tb 0.
P 242A 24
y 12 sina
12
tb 73.
A
85 61
2
12.D 73,
M m
2
85 61
2
,y A sina
12
tb D.
B
12
,P 24,
t 065°F.
61°F.
85°F
M m
2
M m
2
D 6 0.
D 7 0
y A sin1Bt2 D.
B
12
.
24
2
B
P 24,
A
15t 0
0°C 132°F2,
y A sin1Bt2 D
College Algebra & Trignometry—
6121824
t
C
15
15
Figure 5.79
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5-87 Section 5.7 Transformations and Applications of Trigonometric Graphs 589
This gives the graph of Note the brokenline notation
“ ” in Figure 5.81 indicates that certain values along an axis are unused (in this
case, we skipped to and we began scaling the axis with the values needed.
c. As indicated in Figure 5.81, the temperature hits twice, at about 15 and
21 hr after 12:00 noon, or at 3:00
A.M. and 9:00 A.M. Verify by computing
f(15) and f(21).
Now try Exercises 7 through 18
Sinusoidal graphs actually include both sine and cosine graphs, the difference
being that sine graphs begin at the average value, while cosine graphs begin at the
maximum value. Sometimes it’s more advantageous to use one over the other, but
equivalent forms can easily be found. In Example 2, a cosine function is used to model
an animal population that fluctuates sinusoidally due to changes in food supplies.
EXAMPLE 2
Modeling Population Fluctuations Using a Cosine Function
The population of a certain animal species can be modeled by the function
where represents the population in year t.
Use the model to
a. Find the period of the function.
b. Graph the function over one period.
c. Find the maximum and minimum values.
d. Estimate the number of years the population is less than 8000.
Solution
a. Since the period is meaning the population of this
species fluctuates over a 10-yr cycle.
b. Use a reference rectangle ( by units) and the rule of fourths
to locate zeroes and max/min values, then sketch the unshifted graph
With these occur at , 2.5, 5, 7.5, and 10
(see Figure 5.82). Shift this graph upward 9000 units (using an appropriate
scale) to obtain the graph of P(t) shown in Figure 5.83.
t 0P 10,y 1200 cosa
5
tb.
P 102A 2400
P
2
/5
10,B
5
,
P1t2P1t2 1200 cosa
5
tb 9000,
65°
60°2,0°
y 12 sina
12
tb 73.
6121824
t (hours)
F
12
6
12
6
0
y 12 sin t
12
Figure 5.80
y 12 sin t 73
12
12 18
24
6
0
t (hours)
Average value
F
65
75
85
60
70
80
90
(c)
Figure 5.81
WORTHY OF NOTE
Recall from Section 5.5 that
transformations of any
function remain
consistent regardless of the
function f used. For the sine
function, the transformation
is more
commonly written
, and
gives a vertical stretch
or compression, C is a
horizontal shift opposite the
sign, and D is a vertical shift,
as seen in Example 1.
A
y A sin1t C2 D
y af1x h2 k
y f1x2
College Algebra & Trignometry—
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